vol1_pp18-33. 173KB Jun 04 2011 12:06:06 AM

The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 1, pp. 18-33, June 1996.
ISSN 1081-3810. http://gauss.technion.ac.il/iic/ela

ELA

SOME PROPERTIES OF THE Q-ADIC
VANDERMONDE MATRIX
VAIDYANATH MANIy AND ROBERT E. HARTWIGy

Abstract. The Vandermonde and con
uent Vandermonde matrices are of fundamental
signicance in matrix theory. A further generalization of the Vandermonde matrix called the
q-adic coecient matrix was introduced in [V. Mani and R. E. Hartwig, Lin. Algebra Appl.,
to appear]. It was demonstrated there that the q-adic coecient matrix reduces the Bezout
matrix of two polynomials by congruence. This extended the work of Chen, Fuhrman, and
Sansigre among others. In this paper, some important properties of the q-adic coecient
matrix are studied. It is shown that the determinant of this matrix is a product of resultants
(like the Vandermonde matrix). The Wronskian-like block structure of the q-adic coecient
matrix is also explored using a modied denition of the partial derivative operator.

Key words. Vandermonde, Hermite Interpolation, q-adic expansion.
AMS(MOS) subject classication. Primary 15A15, 15A21, 41A10, 12Y05

1. Introduction. Undoubtedly, the Vandermonde matrix is one of the
most important matrices in applied matrix theory. Its generalization to the
con
uent Vandermonde matrix has numerous applications in engineering. This
concept was further generalized to the q -adic Vandermonde matrix in [9], extending the work of Chen [4], Fuhrman [7] and Sansigre [3] among others. In
this paper, we investigate some of the properties of the q -adic Vandermonde
matrix. In particular, we shall evaluate its determinant and analyze its block
structure. The former extends the Vandermonde determinant, while the latter shows a remarkable analogy to the layered Wronskian structure of the
con
uent Vandermonde matrix.
It is well known that the con
uent Vandermonde matrix for the polynomial
(x) =

Ys (x ,  )m ;

i=1

i

i

deg( ) = P mi = n, can be constructed by forming the Taylor expansion of
Received by the editors on 5 December 1995. Final manuscript accepted on 29 May
1996. Handling editor: Daniel Hershkowitz.
y
North Carolina State University, Raleigh, North Carolina 27695-8205, USA
(vmani@iterated.com, hartwig@math.ncsu.edu).
18


ELA
19

Properties of q-adic Vandermonde Matrix

xt in terms of the powers of (x , i), i.e.,

1

x
x2

= 1
= i + 1(x , i )
= 2i + 2i(x , i ) + 1(x , i )2

xt

= ti + tti,1 (x , i) + : : :

..
.

(1)

..
.

xn,1

=



Pn

j =1

n , 1  n,j (x , i )j :
j,1 i

If we set (x , i ) = qi and denote the column containing the rst mi coecients
in the above expansion of xt by [xt](q ;m ) , then
i

[xt ]

(2)

qi;mi ) =

(

6
6
6
6
6 
4

i

ti
tti,1

2

3

..
.

t
t,m +1
mi , 1 i
i

7
7
7
7
7
5

:

mi 1

In case t < mi , 1, we of course replace the elements involving negative powers
of i by zero. Collecting these coordinate columns, we may construct the nmi
matrix
nm (i) as
i

i

h


nm (i)T = [1](q ;m ) [x](q ;m ) : : : [xn,1 ](q ;m ) :
i

i

i

i

i

i

i

In other words,
nm (a) is made
up of rst
mi columns of Caratheodory's n  n
n,1;n,1

. Lastly, the n  n con
uent
binomial matrix
n (a) = kj ak,j
k;j =0
Vandermonde is dened by matrix



(3)

 =
nm1 (1);
nm2 (2); ::;
nm (s ) :
To extend this concept further, we recall the q -adic expansion of a given
polynomial f (x) relative to a monic polynomial q (x). Indeed, the following
lemma from [2] is elementary.
Lemma 1.1. Let q (x) be a monic polynomial of degree l over F. If f (x)
is any polynomial over F, then there exist unique polynomials rj (x) over F, of
degree less than l, such that
i

s

f (x) =

k

X
j =0

rj (x)[q (x)]j

ELA
20

Vaidyanath Mani and Robert E. Hartwig

for some nite non-negative integer k.
Let us now use the above lemma to dene the q -adic coordinate column of
a polynomial. For an arbitrary polynomial f (x) = a0 +a1 x+: : :+ak xk , its standard coordinate column relative to the standard basis BST = f1; x; : : :; xn,1 g
is dened and denoted by

f

a0 3
a1 77

2
6
= [f ](st) = 664

.. 75 :
.

ak

Now for a given pair (q; m) made up of a polynomial q (x) of degree l and a
positive integer m, the q-adic expansion of f (x) may be written as
f (x) = r0(x) + r1(x)q + : : : + rm,1 (x)q m,1 + q m(?);
where ? denotes the (polynomial) quotient after division by q m . The q-adic
coordinate column associated with f (x) for the pair (q; m) is now dened as
3
7
7
7
5

2
r0
6
r1
[f ](q;m) = 664 ..
.

rm ,

;

lm1
where ri is the standard coordinate column for ri (x). It should be noted that
(i) if the degree of f (x) is larger than q (x)m , we simply ignore the quotient
after division by q m .
(ii) the statements q m jf and [f ](q;m) = 0 are equivalent, i.e.,
(4)
[f ](q;m) = 0 , q m jf:
1

(iii) if deg (f ) = k, then [f ](st) is a (k + 1)  1 column vector.
Throughout this paper, we shall denote the ith unit column vector by ei .
1.1. The q-adic Vandermonde matrix. We are now ready for the
construction of the q -adic Vandermonde matrix. In order to repeat the construction given in (1) over elds which are not algebraically closed, let (x)
be a foundation polynomial with the factorization
s
Y
(x) = q m
i=1

i

i

Ps

with deg (qi) = li  1 and deg ( ) = i=1 mi li = n. Again, using the q-adic
expansion of the standard basis elements xt , we have

xt =

mX
i ,1
j =0

rij(t)(x)qi(x)j + qi (x)m (?):
i

ELA
21

Properties of q-adic Vandermonde Matrix

The qi -adic coordinate column [xt](qi;mi ) may now be formed as
2
6
6
t
[x ](qi;mi ) = 666
4

(5)

3
7
7
7
;
7
7
5

rit
rit

( )
0
( )
1

..
.

t
ri;m
i,
( )

1

where r(ijt) is the standard coordinate column for rij(t)(x). Lastly, collecting
coordinate columns, we set
(6)

h

Wi = [1](qi;mi) [x](qi;mi) : : : [xn,1 ](qi;mi )

i

and construct the n  n q -adic Vandermonde matrix W as

li mi n

2
W1 3
6 27
7
W T = 64 W
::: 5

Ws

It is clear that when the qi 's are linear polynomials then, (2) reduces to (6) and
W T to
T . We shall as such refer to the matrix W as the q -adic Vandermonde
matrix. If W is nonsingular, it solves the q -adic interpolation problem for
polynomials as discussed in [9]. Moreover, for any polynomial f (x) with degree
less than n and standard coordinate column [f ](st), it is clear from (5) that
(7)

Wi [f ](st) = [f ](qi;mi );

In other words, Wi acts as a change of basis matrix from the standard to the
qi -adic coordinates. This is a crucial observation which will be used in the

next section.
To analyze W further, we need the relation between the q-adic coordinate
columns of [f ](q;m) and [k f ](q;m). This relation is best handled via the concept
of a polynomial in a hypercompanion matrix which we shall now address.
1.2. Some basic results hypercompanion matrices. If q(x) = a0 +
a1x + : : : + xl is a monic polynomial of degree l, then the hypercompanion
matrix for q (x)m is the m  m block matrix
(8)

2
Lq 0 :
66 N Lq 0
Hqm = 66 : : :
4

0
0

0
0

::
::
:
N Lq
0 N

0
0
0
0

Lq

3
7
7
7
;
7
5

ELA
22

Vaidyanath Mani and Robert E. Hartwig

where N = E1;l = e1 eTl and Lq is the companion matrix of q (x) dened by
2
0 0 0 ::: ,a0 3
6 1 0 0 ::: ,a1
7
6
7
6 0 1 0 ::: ::
7
6
7:
Lq = 6 : : : ::: ::
7
6
4

: : : ::: ,ak,2
0 0 0 ::1 ,ak,1

7
5

The key result which we need is the following lemma dealing with polynomials
of hypercompanion matrices.
Lemma 1.2. Let p(x) be a monic polynomial of degree l and let Hm =
H [p(x)m] be the ml  ml hypercompanion matrix induced by p(x). If f (x) is
any polynomial over F with p-adic expansion
(9) f (x) = 0(x) + 1 (x)p(x) + : : : + m,1 (x)p(x)m,1 + p(x)m(?);
and coordinate columns i , then
2
A0
0
::: 0 3
6 A1
A0
0 77 
m,1 U  ;
(10)f (Hm ) = 664 ..
U;
JU;
:
:
:;
J
7=
.
..
.
.
.
. . 5
.
.

Am,1 Am,2 : : : A0
where A0 = 0 (Lp ), Ai = i (Lp ) + i,1 (Lp + N ) , i,1 (Lp), i = 1; : : :; m , 1,
N =2 E1;l and3 U = [a; H a; : : :; H l,1a] with J = Jm (0)
Il = p(Hm) and
0
7
6 1
7
6
a=6
4

..
.

m,1

7
5

.

Proof. To calculate a polynomial in the hypercompanion matrix Hm =
H [p(x)m], we proceed in two stages. First suppose that f (x) has degree
deg (f ) < deg(p) = l. Now split Hm as Hm = Im
L + Jm(0)
N , and
since
(11)
NLk N = 0; k = 0; 1; : : :; l , 2;

it follows by induction that (L + N )k = Lk + Yk ; k = 0; 1; : : :; l, and that
(12)
Hmk = I
Lk + Jm (0)
Yk ; k = 0; 1; : : :; l
where Yk = Lk,1 N + Lk,2 NL + : : : + NLk,1 and Y0 = 0.
Next, we dene the dierence operator for xed p(x) and f (x) by

f = f (L + N ) , f (L):

ELA
23

Properties of q-adic Vandermonde Matrix

Summing up (12) shows that if deg (f ) < l, then
(13)
f (Hm) = Im
f (L) + Jm (0)

f:
This formula breaks down when deg (f )  l, since (11) does not hold for values
of k  l , 2. But we may proceed in the following way. Suppose f (x) admits
the p-adic expansion
f (x) = 0(x) + 1(x)p(x) + : : : + m,1(x)p(x)m,1 + p(x)m(?):
Now recall from [6] that J = p(Hm) = Jm (0)
Il . Clearly, p(Hm )k = 0 for
k  l. Hence by (13), k (Hm)p(Hm)k has the form
k (Hm)p(Hm)k = Jm (0)k
k (L) + Jm(0)k+1

k ;
for k = 0; 1; : : :; l , 1. Summing this for k = 0; 1; : : :; l , 1, we see that f (Hm )
is block Toeplitz of the form given in (10).
For the
part it suces to show that U agrees with the matrix
2 remaining
3

A0
A1 77. It is well known that for a polynomial
::: 5
Am,1
h(x) = h0 + h1x + : : : + hn,1 xn,1, h(L) = [h Lh : : : Ln,1 h], where L is
any n  n companion matrix. Moreover, (Le ) = L + N; N = E1;n is also a

6
A = 64

companion matrix. From the above two observations, it follows that
A0e1 = k (L)e1 + k,1 (Le)e1 , k,1 (L)e1 = k + k,1 , k,1 = k :
Needless to say, this ensures that Ae1 = a = f (Hm )e1 . Consequently, Hm a =
f (Hm)Hm(e1) = f (Hm)e2. Similarly, Hmk a = f (Hm)ek+1 for k = 0; 1; : : :; l,1
and hence U = [a; Hm a : : : Hml,1 a], completing the proof.
An immediate corollary to Lemma 1.2 gives us the required result relating
the q -adic coordinate column of [f ](q;m) and [k f ](q;m).
Corollary 1.3. Let [f ](q;m) = a for polynomials f (x) and q (x) as in
Lemma 1.2. Then [k f ](q;m) = (Hm )k a and [q k f ](q;m) = J k a, where Hm =
H [qm] and J = Jm(0)
Il.
Proof. From Lemma 1.2, we see that for any polynomial g (x), [g ](q;m) =
g(Hm)e1: Consequently, if g(x) = k f (x), then
[g ](q;m) = g (Hm)e1 = (Hm )k f (Hm )e1 = (Hm )k a:
The remaining result is clear since q (Hm ) = J .
Given the special construction of W , it comes as no surprise that W is
directly related to the hypercompanion matrix. Indeed, since [1](q ;m ) = e1 ,
it is clear from Corollary 1.3 that [k ](q ;m,i) = Hik e1 ; where Hi = H [qim ].
Consequently,
i
h
(14)
Wi = e1 Hie1 : : : Hin,1e1 :
i

i

i

i

ELA
24

Vaidyanath Mani and Robert E. Hartwig

1.3. The E matrix. In order to calculate the determinant of W , our
strategy shall be to block diagonalize W T rst, and then compute the determinant of the resulting block diagonal matrix. In other words, we wish to
nd a suitable non singular matrix E = [E1; E2; : : :; Es] such that W T E =
diag(D1; D2; : : :; Ds), i.e., we want

0 i 6= j
WiEj = D
i i = j;
where Wi is li mi  n and Ej is n  lj mj .
Based on (4) and (7), it suces to take as the columns of Ej , the standard
coordinate columns of suitable polynomials all divisible by qim . In addition, we
want E to be such that the blocks Di = Wi Ei are manageable and det(E ) 6= 0.
A convenient choice is the chain of polynomials
i

j = [xk qj (x)r j (x)] = j (x)[j ; qj j ; : : :; qjm ,1 j ]
j

where j = =qjm , j = [1; x; : : :; xl ,1 ], j = 1; : : :; s, r = 0; : : :; mj , 1 and
k = 0; 1; : : :; lj , 1. Clearly, qim divides every polynomial in j for i 6= j . As
mentioned earlier, Ej is dened by
(15)
j = BST Ej :
From (4), we have
j

j

i

(16)
Wi [xk qj (x)r j (x)](st) = [xk qj (x)r j (x)](q ;m )
and hence for j 6= i, Wi Ej = 0. On the other hand when j = i, we recall
Corollary 1.3, so that (16) reduces to
i

i

Wi [xk qi (x)r i(x)](st) = [kqir i](q ;m ) = Jir Hik a;
where Hi = H [qim ], Ji = Jm
Il and a = [ i ](q ;m ) . Now if [U ] =
[a; Hia; : : :; Hil ,1 a] then an application of Lemma 1.2 yields
i

i

i

i

i

i

i

i

h

i

WiEi = U JU : : : J m ,1 U = i(Hi):
i

We have thus shown the following result.
Lemma 1.4. For the matrices W and E described above
(17)
W T E = diag[ i(Hi )]si=1;
where m i = =qim and Hi = H (qim ) is the hypercompanion matrix associated
with qi .
We close this section with the following remarks.
i

i

i

ELA
25

Properties of q-adic Vandermonde Matrix

Remark 1.5. The blocks Er in E = [E1; E2; : : : ; Es] may be further

described as
(18)

Er

= r (L)



Ili



; qi (L)



Ili



0
0
see (5.4) of [5]. To obtain the above, simply note that
k
r (L)qr (L)



Ilr

0





; :::::; qi (L)mi ,1

h

= qr (L)k r ; L r; : : : ; Ll ,1 r
r

h

Ili

0

i



;

;
i

= [qrk r ](st); [xqrk r ](st); : : : ; [xl ,1 qrk r ](st) ;
in which L = L and [ r ](st) = r are respectively the companion matrix and
the standard coordinate column of (x). Since n  deg ( r ) + lr , we see that
for k = 0; 1; : : :; lr , 1, Lk r merely shifts the nonzero entries in r without
losing any of them.
Remark 1.6. In the next section we shall explicitly calculate the determinant of the E matrix. It will be shown that det(E ) is nonzero, and hence
E is nonsingular, if and only if the polynomials qi are pairwise coprime, i.e,
gcd(qi; qj ) = 1 for i 6= j .
Remark 1.7. It was shown in [9] that the polynomials  form a Jacobson
chain basis for the space Fn,1 [x] of polynomials of degree less than n with
respect to a shift operator S . Hence, when the qi 's are pairwise coprime, E is
a change of basis matrix from the standard to a chain basis denoted by BCH .
The reader is referred to [9] for more details.
Let us now capitalize on (17) and simultaneously calculate the determinant
of E as well as that of W .
2. Determinants. It is well knownQthat if
is the con
uent Vandermonde matrix for the polynomial  (x) = si=1 (x , i)m , then
Y
det(
) =
(i , j )m m ;
r

i

i

j

j